ON THE DYNAMICS OF $x_{n+1}=\frac{a+x_{n-1}x_{n-k}}{x_{n-1}+x_{n-k}}$

Abstract

In this paper, we investigate the behavior of solutions of the difference equation $$x_{n+1}=\frac{a+x_{n-1}x_{n-k}}{x_{n-1}+x_{n-k}},\;n=0,1,2,{\ldots}$$ where $k{\in}\{1,2\}$, $a{\geq}0$, and $x_{-j}$ > 0, $j=0,1,{\ldots},k$.

1. Introduction

Difference equations appear as natural descriptions of observed evolution phenomena because most measurements of time evolving variables are discrete and as such these equations are in their own right important mathematical models. More importantly, difference equations also appear in the study of discretization methods for differential equations. Several results in the theory of difference equations have been obtained as more or less natural discrete analogues of corresponding results of differential equations.

Recently there has been a lot of interest in studying the global attractivity, boundedness character, periodicity and the solution form of nonlinear difference equations. For example,

Abu-Saris et al.[1] investigated the asymptotic stability of the difference equation

For other related results([2-16]).

In this paper, we investigate the behavior of solutions of the difference equation

where k ∈ {1,2}, a ≥ 0, and x-j > 0, j = 0, 1, ..., k.

We need the following definitions.

Definition 1.1. Let I be an interval of real numbers and let

be a continuously differentiable function. Consider the difference equation

with x−k, x−k+1, ..., x0 ∈ I. Let be the equilibrium point of Eq.(1.2). The linearized equation of Eq.(1.2) about the equilibrium point is

where The characteristic equation of Eq.(1.3) is

(i) The equilibrium point of Eq.(1.2) is locally stable if for every ϵ > 0, there exists δ > 0 such that for all x−k, x−k+1, ..., x−1,x0 ∈ I with

we have

(ii) The equilibrium point of Eq.(1.2) is locally asymptotically stable if is locally stable and there exists γ > 0, such that for all x−k, x−k+1, ..., x−1,x0 ∈ I with

we have

(iii) The equilibrium point of Eq.(1.2) is global attractor if for all x−k, x−k+1, ..., x−1,x0 ∈ I we have

(iv) The equilibrium point of Eq.(1.2) is globally asymptotically stable if is locally stable, and is also a global attractor of Eq.(1.2).

(v) The equilibrium point of Eq.(1.2) is unstable if is not locally stable.

Definition 1.2. A positive semicycle of of Eq.(1.2) consists of a ‘string’ of terms {xl, xl+1, ..., xm}, all greater than or equal o , with l ≥ -k and m ≤∞ and such that either l = -k or l > -k and and either m = ∞ or m < ∞ and

A negative semicycle of of Eq.(1.2) consists of a ‘string’ of terms {xl, xl+1, ..., xm}, all less than , with l ≥ -k and m ≤ ∞ and such that either l = -k or l > -k and and either m = ∞ or m < ∞ and .

Definition 1.3. A solution of Eq.(1.2) is called nonoscillatory if there exists N ≥ -k such that either

and it is called oscillatory if it is not nonoscillatory.

We need the following theorems.

Theorem 1.4 ([16]). (i) If all roots of Eq.(1.4) have absolute value less than one, then the equilibrium point of Eq.(1.2) is locally asymptotically stable.

(ii) If at least one of the roots of Eq.(1.4) has absolute value greater than one, then is unstable.

The equilibrium point of Eq.(1.2) is called a saddle point if Eq.(1.4) has roots both inside and outside the unit disk.

Theorem 1.5 ([16]). Assume that p1, p2, ..., pk ∈ ℝ and k ∈ {1, 2, ...}. Then

is a sufficient condition for the asymptotic stability of the difference equation

 

2. Behavior of solutions of Eq.(1.1) when k = 1 and a = 0.

In this section we give the closed form of solutions of Eq.(1.1) when k = 1 and a = 0.

In this case the difference equation (1.1) becomes

with positive initial conditions x-1 and x0.

Eq. (2.1) is linear which have the solution

 

3. Behavior of solutions of Eq.(1.1) when k = 2 and a = 0.

In this section we investigate the behavior of solutions of Eq.(1.1) when k = 2 and a = 0.

In this case the difference equation (1.1) becomes

with positive initial conditions x-2, x-1 and x0.

Eq.(3.1) has a unique equilibrium point

Theorem 3.1. The equilibrium point of Eq.(3.1) is locally asymptotically

Proof. Since the linearized equation of Eq.(3.1) about the equilibrium point can be written in the following form

then the proof follows immediately from Theorem B.

Theorem 3.2. The equilibrium point of Eq.(3.1) is globally asymptotically stable.

Proof. From Eq.(3.1) it is easy to show that xn+1 < xn−1 for all n ≥ 0 and so the even terms converge to a limit (say L1 ≥ 0) and the odd terms converge to a limit (say L2 ≥ 0). Then

which implies that L1 = L2 = 0, and the proof is complete.

 

4. Behavior of solutions of Eq.(1.1) when k = 1 and a > 0.

In this section we investigate the behavior of solutions of Eq.(1.1) when k = 1 and a > 0.

In this case the difference equation (1.1) becomes

with positive initial conditions x−1 and x0.

The change of variables reduces Eq.(4.1) to the difference equation

Eq.(4.2) has a unique positive equilibrium point

Theorem 4.1. The equilibrium point of Eq.(4.2) is locally asymptotically stable.

Proof. The linearized equation of Eq.(4.2) about the equilibrium point is

and so, the characteristic equation of Eq.(4.2) about the equilibrium point is

which implies that |λ1| = |λ2| = 0 < 1. Hence, the proof is complete.

Theorem 4.2. The equilibrium point of Eq.(4.2) is globally asymptotically stable.

Proof. Since for all n ≥ 0, then we have yn ≥ 1 for all n ≥ 1.

Furthermore for all n ≥ 2. So the even terms converge to a limit (say L1 ≥ 0) and the odd terms converge to a limit (say L2 ≥ 0). Then

which implies that L1 = L2 = 1. Thus, the proof is complete.

 

5. Behavior of solutions of Eq.(1.1) when k = 2 and a > 0.

In this section we investigate the behavior of solutions of Eq.(1.1) when k = 2 and a > 0.

In this case the difference equation (1.1) becomes

with positive initial conditions x−2, x−1 and x0.

The change of variables reduces Eq.(5.1) to the difference equation

Eq.(5.2) has two equilibrium points

Theorem 5.1. The equilibrium point of Eq.(5.2) is unstable equilibrium point.

Proof. The linearized equation of Eq.(5.2) about the equilibrium point is

and so, the characteristic equation of Eq.(5.2) about the equilibrium point is

is clear that f (λ) has a root in the interval (1,∞), and so, is an unstable equilibrium point. This completes the proof.

Theorem 5.2. The equilibrium point of Eq.(5.2) is locally asymptotically stable.

Proof. The linearized equation of Eq.(5.2) about the equilibrium point is

and so, the characteristic equation of Eq.(5.2) about the equilibrium point is

which implies that |λ1| = |λ2| = |λ3| = 0 < 1, from which the proof is complete.

Lemma 5.3. The following identities are true

Proof. (i) for n ≥ 0.

(ii) for n ≥ 0.

(iii) for n ≥ 0.

(iv)

(v)

Then, the proof is complete.

Theorem 5.4. Let be a solution of Eq.(5.2), then the following statements are true

(i) If for some n0 ∈ {-1, 0, 1, 2, ...}, then for all n ≥ n0 + 2.

Also if then for all n ≥ 3.

(ii) If for some n0 ∈ {-2,-1, 0, 1, 2, ...}, then for all n ∈ n0.

(iii) If (i) and (ii) are not satisfied, then oscillates about , with positive semicycles of length at most three, and negative semicycles of length at most two.

Proof. (i) Let for some n0 ∈ {-1, 0, 1, 2, ...}, then from Eq.(5.3) we have for all n ≥ n0 + 2.

If then from Eq.(5.3) we have which implies that for all n ≥ 3.

(ii) Let for some n0 ∈ {-2,-1, 0, 1, 2, ...}, then from Eq.(5.3) we have for all n ∈ n0.

(iii) Suppose without loss of generality that there exists n0 ∈ {-2,-1, 0, 1, 2, ...}, such that Then from Eq.(5.3) we have The proofs of the other possibilities are similar, and will be omitted.

Theorem 5.5. The equilibrium point of Eq.(5.2) is globally asymptotically stable.

Proof. We proved that of Eq.(5.2) is locally asymptotically stable, and so it suffices to show that limn→∞ zn = 1. If there exists n0 ∈ {-2,-1, 0, 1, 2, ...}g, such that then from Theorem 5.4 we have limn→∞ zn = 1. Also, if then by Theorem 5.4 we have for all n ≥ -2, and from Eq.(5.4), we have zn+1 > zn-1, for n ≥ 0. So the sequences are increasing and bounded, which implies that the even terms converge to a limit (say M1 > 0) and the odd terms converge to a limit (say M2 > 0). Then

which implies that M1 = M2 = 1.

Now, Suppose that then from Eqs.(5.4) - (5.7) we have the following results

The sequence is decreasing and bounded, and so converges to a limit (say L0 > 0).

The sequence is increasing and bounded, and so converges to a limit (say L1 > 0).

The sequence is increasing and bounded, and so converges to a limit (say L2 > 0).

The sequence is decreasing and bounded, and so converges to a limit (say L3 > 0).

The sequence is increasing and bounded, and so converges to a limit (say L4 > 0).

The sequence is decreasing and bounded, and so converges to a limit (say L5 > 0).

The sequence is decreasing and bounded, and so converges to a limit (say L6 > 0).

So we have from Eq.(5.2) that

The solution of this system is either Li = -1, i = 0, 1, ...6, or Li = 0, i = 0, 1, ...6, or Li = 1, i = 0, 1, ...6. Since Li > 0, i = 0, 1, ...6 , we have limn→∞ zn = 1.

The proofs for the other cases are as follows.

or are similar to the proof of the last case, and will be omitted. Therefore the proof is complete.